01-434 Ostap Hryniv and Dima Ioffe
Self-avoiding polygons: Sharp asymptotics of canonical partition functions under the fixed area constraint (726K, ps) Nov 26, 01
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Abstract. We study the self-avoiding polygons (SAP) connecting the vertical and the horizontal semi-axes of the positive quadrant of $\mathbb{Z}^2$. For a fixed $\beta>0$, assign to each such polygon $\omega$ the weight $\exp\{-\beta|\omega|\}$, $|\omega|$ denoting the length of $\omega$, and consider the sum $Z_{Q,+}$ of these weights for all SAP enclosing area $Q>0$. We study the statistical properties of such SAP and, in particular, derive the exact asymptotics for the partition function $Z_{Q,+}$ as $Q\to\infty$. The results are valid for any $\beta >\beta_c$, $\beta_c$ being the critical value for the 2D self-avoiding walks.

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